Centroid is a geometric point that represents the center of mass or the average position of all points in a shape or object, often used in mathematics, physics, and engineering for various analytical purposes. Centroid always lies within the figure and is not only related to triangles; it can be determined for every geometric figure as well.

In this article, we will explore the concept of the centroid in detail, including the centroid of triangles as well as centroid of various geometric shapes such as triangles, quadrilaterals, polygons, as well as circles. Additionally, we will learn about the formula to calculate the centroid of a triangle using the coordinates of its vertices.

What is Centroid?

In geometry, a centroid is a point defined as the arithmetic mean position of all the points in a shape. It is often referred to as the “center of mass” or the “center of gravity” of a geometric object. The centroid depends on the distribution of mass or points within the object and is a useful concept in various branches of geometry, physics, and engineering.

Centroid Definition

For a two-dimensional object, such as a triangle or a polygon, the centroid is the point where the medians intersect.

The median of triangle is a line segment connecting a vertex to the midpoint of the opposite side. The centroid divides each median into two segments, with one segment twice as long as the other.

For a three-dimensional object, like a solid with uniform density, the centroid is the point where the medians (or analogously, the lines connecting opposite faces’ centroids) intersect.

Properties of Centroid

The properties of centroid are mentioned below:

• Centroid is the center point of object.
• Centroid always lies inside the object.
• Centroid is center point of gravity.
• Centroid never lies outside the object.
• For different objects with different shape , centroid are calculated separately.

Centroid of a Triangle

Similarly, we can derive centroid of more geometric figure by calculating mid point that lies inside the figure. In mathematics centroid is mainly concerned with triangles. Centroid is point inside triangle , where all three medians of triangle intersect. In further section we will derive the formula of centroid of triangle and discuss some problems based on it.

Note: Median of triangle is defined as the line joining the vertex of triangle with the opposite side and bisecting the opposite side.

Centroid Definition in Triangle

Centroid is the point of triangle where all medians of triangle meet.

In other words , the point of intersection medians of triangle is known as centroid of triangle. The centroid of a triangle always lies inside the triangle. Centroid is also called as geometric center of triangle.

In the given triangle ABC, the coordinates of triangle are (x₁ , y₁), (x₂ , y₂), (x₃ , y₃) and the medians from all three vertex A, B and C meet at point G which is centroid of the triangle.

Properties of Centroid in Triangles

Some of the properties of the point named as centroid of triangle

• Centroid is the meeting point of all the three medians of triangle.
• Centroid lies inside the triangle.
• Centroid divides the median in ratio of 2:1.
• Median bisects the opposite side of vertex .
• Centroid never lies outside the triangle.

Centroid Formula for Triangle

The centroid of a triangle can be calculated by using centroid formula. If the vertices of triangle are in the form of (x₁ , y₁) , (x_{2 ,} y₂) and (x₃ , y₃) then centroid of triangle can be define as:

Centroid of Triangle; (x , y) = (x₁+x₂+x₃/3, y₁+y₂+y₃/3)

Where,

• x₁ , x₂ and x₃ are x coordinates of vertices of triangle and
• y₁ , y₂ and y₃ are y coordinates of vertices of triangle.

Centroid in Plane Figures

Centroid for some common plane figures are discussed as follows:

Centroid of Quadrilateral

Centroid of a quadrilateral can be easily calculated by moving half the distance in length direction and half distance in breadth direction. As quadrilateral is a two dimensional figure it has two dimensions of length and breadth only. The mid point inside quadrilateral is centroid point.

Centroid of Rectangle

Rectangle is a solid closed structure bounded by four lines. The opposites sides of rectangle are equal and parallel. Centroid of rectangle lies in the mid of triangle at half distance of length and half distance of breadth.

Suppose if length of rectangle is L unit and breadth of rectangle is B unit then centroid of rectangle will lie on point (L/2 , B/2).

Centroid of Rectangle

Centroid of Square

Square is closed geometrical figure bounded by four lines. Unlike rectangles, all the sides of square are equal and all interior angles of square is equal to 90 deg. Centroid of square lies at the mid point of square. All sides of square are equal so suppose if side of square is S unit then centroid of square lie on point ( S/2 , S/2).

Centroid of Square

Centroid of Polygon

Centroid of a polygon is calculated by taking integration of all points on polygon and dividing it by total number of sides present in it. Centroid is calculated by using formula,

Centroid of Polygon = 1/n(integration of all points in polygon)

Centroid of Circle

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed shape, two-dimensional shape, curved shape. For circle centroid overlap on center of circle. If diameter of circle is D then centroid lies on point D/2.

Centroid of Circle

Centroid in Solid Figures

Centroid in a solid figure is point of center of gravity. As centroid is center point of object.

Formula for calculating the centroid in solid figures :

C_x = ∫x dV/V , C_y = ∫y dV/V and C_z = ∫z dV/V

Where,

• The perpendicular distance in the x direction from the yz-plane to the centroid is C_x,
• The perpendicular distance in the y direction from the zx-plane to the centroid is C_y,
• The perpendicular distance in the z direction from the xy-plane to the centroid is C_z, and
• The coordinates of the centroid are (C_x , C_y , C_z).

Centroid Vs Circumcenter Vs Incenter

The key difference between centroid, circumcenter and incenter are listed in the following table: